A few things to note so you don't waste your time and money.
Bluegrass and Ryegrass are not shade-loving and I wouldn't recommend them for your application being half-wooded and semi-shaded. You'll want a cool season, shade-tolerant grass like fescue. Personally, I'd go with a tall fescue.
That being said, it's never as easy as "throw it and let it grow".
Preferably, you'd get a soil test and make up for any nutrient deficiencies with fertilizers but that's probably more than one is willing to do for a yard their dogs run around in. In order to have the best chance to establish and maintain a nice lawn is to first,
dethatch. If you've never had it done, or your lawn (or whatever is left of it) is spongy, go ahead and dethatch it and remvoe any dead grass and/or roots to allow nutrients, air, and water to get to the roots .
Next, you want to
aerate. Rent (if you don't have one) a nice plug aerator. This thing sticks a hollow metal plug into the ground and pulls out thumb-sized plugs of dirt. This allows the soil to become less compacted and allows immediate entry of nutrients, air, and water to the root system.
Next, apply
starter fertilizer and I like to apply Milorganite (an organic fertilizer with Iron). Be sure you apply it per the instructions on the bag. Over applying does not help and can actually hurt the lawn (especially applying too much synthetic fert like the starter) and to a larger extent, can run off and enter our waterways.
Now, lay down a layer of
topsoil. You'll only need about 1/4" (and fill in the holes made by the plug aerator). When that's done,
apply the seed with a rotary or drop spreader (or by hand if the area is small enough). When the seed is laid down, be sure to use a
roller (one of the large ones you fill up with water) and press the seeds into the soil for good seed-to-soil contact.
Immediately
water the lawn. You'll want to water at least 1" every 3 days (split it between morning and evening) or so until the seed starts to germinate (usually, that takes about 21 days, give or take. Could be longer with shade). The best way to know how much you;re watering is to use the "tuna can trick". A tuna can is 1" deep, so set it in the lawn, run your sprinklers, and figure out how long it takes to water in an inch. Mind you, this can take some time depending on your sprinkler setup and lawn size. For example, my lawn is covered with 3 sprinklers and it takes 3 hours to lay down 1" of water. When seed germinates, cut back watering to 1" per week. I like to do (2) waterings at 1/2" during the week.
New grass is delicate, so a ton of traffic on new grass could kill it, so I'd get some temporary fencing and fence off certain areas to allow the grass to grow. Let the grass get to be 3-4" before mowing the first time and only mow a max of 1/3 the height off the grass at a time. You may want to get one area established, wait until Fall and do the same process over again for the other area.
After 3 mowings, apply Milorganite every other month during the growing season (remember that fesues are cool season grasses, so don't apply in the super hot summer months as the grass is dormant). I typically do it around Memorial Day, Independence Day, Labor Day, and Thanksgiving. Milorganite, as I said, is organic and contains Iron. Iron is what gives grass that deep green, almost blue hue and while you can't overfertilize with Milo (it won't burn your lawn like synthetic ferts), you will waste money, so apply per the instructions on the bag. In MD it's 36lbs for a 6,000 sq ft. lawn so it goes a long way.
Now that you're completely bored out of your mind thinking "I'm not doing this ####", you need to know when to overseed. While fall is the best time to overseed with cool season grasses (it gives the roots a chance to establish before the hot summer months), it certainly can be done in the Spring. You'll want to be sure no frost will happen and that soil temps are around 55*-60* with air temps around 70*.
Also, don't apply any pre-emergent like weed killer/preventer or crabgrass preventer until at least your 3rd mowing.
UMD Extension Office has a good wed site on overseeding:
https://extension.umd.edu/hgic/faqs-lawns-overseeding-and-renovation-spring#recently
http://extension.umd.edu/sites/exte...Lawn Establishment Renovation Overseeding.pdf
Milorganite info (you can find this at any home store. Home Depot, ACE Hardware, etc.)
https://www.milorganite.com/
All else fails, try sod.